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Structural Analyses of Segmental Lining – Coupled Beam andSpring Analyses Versus 3D-FEM Calculations with Shell ElementsC. Klappers, F. Grübl, B. OstermeierPSP Consulting Engineers for Tunnelling and Foundation Engineering, Munich, GermanyABSTRACTIn contrast to the inner lining of a NATM tunnel the lining of a TBM driven tunnel consists of singleprecast concrete segments which are articulated or coupled at the longitudinal and circumferentialjoints. Therefore not only the characteristics of the concrete segments influence the structure but alsothe mechanical and geometrical characteristics of the joints strongly affect the structural behaviour ofthe tunnel lining. For the simulation of these joints within the tunnel lining different calculationmethods are known. In the following it is shown how the behaviour of the joints can be modelled in an appropriateway. Different calculation methods with beam and spring models and 3D-FEM models are comparedand discussed. It can be seen, that for the structural design of the segments for regular casescalculations with special beam and spring models are sufficient whereas 3D-FEM calculations arenecessary when the spatial bearing behaviour of the lining with respect to the bearing behaviour of thejoints needs to be considered.1. INTRODUCTIONCurrently beam and spring models (BSM) analysis with coupled, hinged rings can be considered asstate of the art model for the structural design of a segmental lining. However, in special cases such asopenings in the lining for cross passages, BSM analysis do not provide reliable results since thestructural behaviour of the tunnel lining in longitudinal direction, the deformation of the lining due tothe rotation in the longitudinal joints and the relative displacement in the circumferential joints have tobe taken into account. These effects can besimulated with 3D-FEM calculations withbedded shell elements connected with non-linear springs, representing the rotationalstiffness of the concrete hinges in thelongitudinal joints and the coupling of thesegmental rings in the circumferential joints. The different calculation approaches forthe structural design of a segmental lining aredescribed and for a typical configuration of asegmental lining the results of BSM analysiswith coupled, hinged rings are compared withthe results of 3D-FEM calculations. Figure 1. Segmental lining 1

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2. STRUCTURAL DESIGN FOR A SEGMENTAL LINING WITH BEAM AND SPRINGANALYSESAll calculations mentioned in this paper base on a referencetunnel with a system radius of 5.1 m, 40 cm segment thickness,2 m ring length, oedometric modulus of 150 MPA, vertical loadof 250 kPa and Ko=0.6. Each ring is built of 6 segments.Modells are given by two rings in general (ring 1 and ring 2). Two systems are examined. At system I ring 1 has nohinge at the crown and ring 2 is rotated by half a segmentwhich means that there is a hinge at the crown. This is the mostunfavourable configuration in terms of the bending moment atthe crown. At system II all hinges are rotated by 15° comparedto system I.2.1 Different structural systems Figure 2. Ring configurationsFirst of all it has to be differentiated between coupled or uncoupled segmental rings. A lining builtwith straight longitudinal joints behaves as uncoupled hinged ring, whereas in systems built withstaggered joints the rings interact and the distribution of the internal forces is changing. There are a lot of different structural systems known in the design practice to calculate theinternal force within the tunnel lining. The most simple one is to use a rigid bedded ring. This modeldoes not take the behaviour of the joints into account. For an uncoupled system of hinged rings theestimated bending moments are too high and should give conservative results. Sir Allan Muir-Wood(1975) developed a very easy to use empirical formula to estimate the effects of the longitudinal jointsof uncoupled rings in a calculation with a homogenous rigid ring by reducing the bending stiffness ofthe lining. The maximum bending moments calculated with this approach are quite close to themaximum bending moment calculated for a hinged uncoupled ring. For coupled rings these momentsare mostly to small, especially with a configuration like system I. However, this approach is quiteuseful to get a first idea of the forces in the lining. To calculate the internal forces of a segmental lining with staggered joints in a proper way it isessential to simulate the coupling in the circumferential joints. Therefore bedded BSM analyses withcoupled, hinged rings are very common for the structural design of a segmental lining. In all of the following calculations the beams are bedded with non linear radial springs which donot allow tension forces. The assumptions for the behaviour of the joints are done for planelongitudinal and circumferential joints, because in many cases the use of tongue and groove or othertypes of mechanical coupling is deemed to be not necessary or useful .2.2 Bedded beam and spring model analysis with coupled, hinged ringsAs the characteristics of the joints are essential for the structural behaviour of the system themechanical properties of these joints have to be simulatedin an appropriate way. 200 Longitudinal joints: For the determination of the 150rotational stiffness of the longitudinal joints usually the C m [MNm/rad] 100formulas from Janssen (1983) based on the investigations 50of Leonhardt and Reimann (1966) for the resistance 0against rotation and bending of concrete hinges are used. -0,15 -0,1 -0,05 -50 0 0,05 0,1 0,15As long as the joint is completely compressed the M [MNm]rotational stiffness cm is constant and could be described E ⋅ b² Figure 3. Relation of bending momentas cm= . It depends only on the young’s and rotation stiffnes 12modulus E and the width b of the contact zone. If this 2

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bending moment exceeds the boundary bending moment Mbou < N . b / 6 the joint is opening like abird’s mouth. From this point the rotational stiffness depends on the normal forces N and the bendingmoment M and is described as 9⋅ E c M= ⋅ ( 2 ⋅ M − N ⋅ b)³ (1) 32 ⋅ N ³ ⋅ b For the implementation of this behaviour the non linear rotational springs should be able to fulfilthe above mentioned relationship between bending moment and rotation stiffness. It is not necessaryto define a yielding moment because the spring becomes extremely soft if the moment increases tomore than about 80 % of the maximum moment. If only a linear rotational spring with the definition ofa yielding moment is used the estimation of behaviour of the joint seems to be too rough. Circumferantial joints: The coupling of the rings is simulated by lateral springs. In literature thereis not very much published about the modeling of the coupling between the rings. Usually thecoupling of the rings is simulated by using non linear lateral springs which represent the shearstiffness and the maximum bearing capacity of the coupling. When using a plane joint with plywoodhardboards the spring stiffness is given by the shear stiffness of the plywood c=, where G isrepresenting shear modulus, A is the area of hardboard and d is the thickness of hardboard. Even without a mechanical coupling the rings are coupled by friction between plywood andconcrete. This is caused by forces in the circumferential joint due to the influence of the hydraulicshoving rams of the TBM. The value of the frictional coefficient µ is hard to define and is subject ofdiscussions. At laboratory tests which were undertaken for the 4th Elbtunnel Hamburg from STUVA(1996) µ =0.25-0.3 was discovered. Gijsbers and Hordijk (1997) did similar tests for tunnel projects inthe Netherlands. For plywood hardboards they found µ =0.4-0,7 as friction coefficient. After reachingthe maximum force the residual friction coefficient decreased to µ =0.3-0.55. The minimal values forµ were found for normal stresses of about 35 MPA at the hardboards and maximum values for normalstresses of about 12 MPA. Because of the limited compressive WI NGRAF ( V13. 61 -2 1) 1 5 .10 .2 005 PSP Bera tend e Inge nieu re 00 6.strength of concrete normally the area of the hardboard will bechosen big enough that the normal stress at the hardboards will be 00 4.less than 20 MPA. Approximately they will be around 10 an 20 .00MPA. All these tests were done in laboratories with unbedded 2concrete segments where the segments could move independently 0.00from each other. Due to the grouting of the tail gap and thesurrounding ground the deformation of the segments is harmonizedin real conditions on site. -2 -4.00 .00 In the structural analysis the radial springs which simulate thebedding of the rings can also deform independently. Therefore the -6.00 4 0 .0 2. 00 0.00 - 2.00 -4.00 meffect of harmonized deformation has to be considered when Y Str uktur M 1 : 35 X X * 0.819 Z Y * 0 9 .9 6 Z * 0.581 Spring- Be am cou pled R ng Rsy s i =5.1 file :f ull _coup_ V40choosing the frictional coefficient for the coupling springs. Figure 4. Structural system ofBecause of the above mentioned matters taking µ=0,5 into account the coupled spring beam modelseems to be a reasonable value. It will be used in the followingcalculations. For structural final design the value of µ should be varied. Within the analysis themaximum bearing capacity of the lateral springs depends on the chosen frictional coefficient and theapplied shoving forces. The whole system of the model for the BSM analysis consists of two half rings(with respect to the ring length) coupled with the above mentioned lateral springs. 3

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Table 1. Results for different structural systems system I system IIStructural system rigid ring Muir- uncoupled uncoupled coupled coupled uncoupled uncoupled coupled coupled Wood ring ring 1 ring 2 ring 1 ring 2 ring 1 ring 2 ring 1 ring 2max bending 157 132 150 95 206 115 131 122 178 152moment [kNm/m]percentage 119% 100% 114% 72% 156% 87% 99% 92% 135% 115%max settlement at 9 9,9 9 11,6 9,5 9,6 9,5 9,6 9,3 9,3crown [mm]percentage 91% 100% 91% 117% 96% 97% 96% 97% 94% 94%As table 1 shows the calculation with a rigid ring does not give the maximum bending moment. Thebending moments for the coupled rings are always higher. The calculation with the reduced stiffnessaccording to Muir-Wood fits very well to the uncoupled calculations of system II. The coupledcalculations show that ring 1 of systems I behaves much stiffer than ring 2 which causes a loadtransfer from ring 2 to ring 1. This leads to a much higher bending moment at the crown of ring 1.These results demonstrate that for the given loads the ring configuration of system II is morefavourable for the design of the lining. The coupling of the rings reduces the deformation, butincreases the bending moments especially for the “stiffer” ring. From this calculation it can be seenthat for the final design at least for the critical load cases, BSM analyses with coupled, hinged ringsshall be done. With models which are more simple the bending moments might be underestimated.3. CALCULATION WITH A 3D-FINITE-ELEMENT-METHOD (FEM) MODELIn comparison to calculations mentioned in chapter 2 also calculations with a 3D-FEM-program(prepared by SOFiSTiK) were done to check the quality of the results from the BSM analyses.3.1 Modelling of the structureFor the 3D-FEM calculations the tunnel was modelled by a sufficient number of complete rings. Thering configuration is taken as described above in system I. The segments are modelled with plane 4-node shell-elements with a non-conforming formulation. These elements can be bedded in radial andtangential direction. For the bedding non linear effects like failure, yielding and friction can bedefined. Each segment consists of 5 elements in longitudinal direction an 18 elements in tangentialdirection which means 540 elements per ring. At the longitudinal joints the adjacent segments arecoupled with 6 rotational springs. In the circumferential joints the segments are coupled with 3 lateral springs per hardboard which means 72 springs per joint. The mechanical, non-linear properties of the different springs are the same as for the BSM analysis described in chapter 2.2. Since the maximum possible coupling forces depend on the shoving forces of the TBM the calculations were done for a variety of total shoving force between 40 to 5 MN. Figure 5. 3D-FEM-structure 3.2 Comparison of the results of the spring beam model and the 3D-FEM Model With the 3D model coupled and uncoupled systems were calculated. In the figure 6 the effects of 4 uncoupled coupled Figure 6. Deformed structures (scaled up)

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the coupling of the rings are obvious. At the uncoupled system each ring deforms independently and atthe coupled system the deformation of the rings is harmonized. BSM with coupled rings 3D-FEM Structural system uncoupled uncoupled coupled coupled uncoupled uncoupled coupled coupled ring 1 ring 2 ring 1 ring 2 ring 1 ring 2 ring 1 ring 2crown bending 150 95 206 82 155 95 201 82moment [kNm/m]max settlement at 9 11,6 9,5 9,6 9,1 11,2 9,1 9,3crown [mm]Table 2. Comparison of the results of the beam and spring and the 3D-FEM model A comparison of BSM and 3D-FEM model shows, 250that the calculated bending moments of both models are in 230 crown bending moment [kNm]a similar range and deformations differ only slightly. The 210 190deviation of the bending moments calculated with various 170 BSM ring 1total shoving forces is only about 5%. This is because the 150 BSM ring 2coupling forces which are necessary to harmonize the 3D-FEM ring 1 130 3D-FEM ring 2deformation of the rings are very small. If a total shoving 110 90force of more than about 5 MN is applied to the system it 70behaves like the rings were fully coupled. The applied 50 0 5 10 15 20 25 30 35 40shoving force will become more effective to the system if advance force [MN]for example the loads are not equally distributed. 12 For usual cases where the loads and the structure BSM ring 1does not change in longitudinal direction the three- crown settlement [mm] 11 BSM ring 2 3D-FEM ring 1dimensional structural behaviour of the segments has no 3D-FEM ring 2significant influence to the system. That means for this 10kind of load configurations 3D-FEM calculations are not 9necessary. For special cases like openings in the lining,different loads on the rings (e.g. swelling only in partial 8areas), varying bedding conditions for the rings (e.g. if the 0 5 10 15 20 25 30 35 40 advance force [MN]grouting of the tail gap was not done properly at one ring)or other special cases only with 3D-FEM calculations the Figure 7. Bending moment andinternal forces and deformations of the lining can be crown settlementpredicted in a serious way.3.3 Segmental lining with an opening and a temporary bracingVery often the segmental lining has to be opened to build crosspassages between two tubes. During the advance of the passagetunnel it is usual to install a steel framework at the runningtunnel before opening the window. The bearing behaviour ofsuch a structure with a slender steel frame around the openingwas analysed with the 3D-FEM model. Figure 8. Deformed structure 5

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The steel framework is build of rigid beam elements. The horizontal beams are connected to the segments with hinges. The stems are connected to the segments with springs which can only transfer compression forces. With 25 250 respect to the excavation of 230 crown bending moment [kNm] 210 the cross passage, the bedding 20 190 deformation [mm] stiffness around the window isAUTHOR : PSP Beratende Ingenieure 80686 München 15 170PROGRAM : WINGRAF VERSION 13.61-21 (c) SOFiSTiK AG BSM ring 1 reduced and the maximum crown settlement 150 BSM ring 2PROJECT : 3D coupled Ring Rsys=5.1 file:3D_opend_coup_V15 ASB NO. : DATE : 3D-FEM ring 1 10 23.10.2005 differential 130 3D-FEM ring 2 bedding stress is limited to the deformation 110 uniaxial compressive strength 5 90 70 of the surrounding ground. 0 50 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 advance force [MN] advance force [MN] -6.00 Figure 9. Bending moments and deformations of opened ring These calculations show that if the total of shoving forces become smaller than 20 MN combined with reduced possible -4.00 coupling forces the maximum bending moments and the deformations start to increase rapidly. Especially at the invert the bending moment increases about 80 % and the differential radial deformation at the circumferential joints becomes more then 5 -2.00 mm. It can also be seen that with the chosen kind of bracing a minimum coupling between the rings is needed. If the shoving forces become less than 5 MN the investigated system starts to 0.00 become unstable. With simulations like this it is possible to calculate the bearing capacity of the opened lining. It is also possible to define a minimum shoving force which has to be used during the shoving of the tunnel at the area of the cross passage or maybe to decide that another kind of bracing is necessary. 2.00 Figure 10. Stress distribution around the opening 4.00 4. CONCLUSIONS From the shown calculations it can be seen that the structural behaviour of the joints must be taken 6.00 into account within the structural analysis of the segmental lining. For normal load cases beam and spring analyses with coupled hinged rings are sufficient. In special cases were the 3D bearing behaviour of the whole tunnel has to be considered 2.00 FEM calculations with bedded shell elements give a 4.00 6.00 8.00 10.00 12.00 14.00 m good impression of the internal forces and the Sector of system M 1 : 75 deformations of the system. For all types of calculations X Z Plane Principal stresses in Nodes, nonlinear Loadcase 1 GEBIRGSDRUCK+QUELLDRUCK, 1 cm 3D = 7.81 MPa Y += -= (Min=-18.0) (Max=5.97)PART : the behaviour of the joints has to be modelled in a ARCHIV NOBLOCK :DETAIL : proper way, because these joints will highly affect the results. The possible minimum and maximum coupling forces have to be taken into account and a parametric study with a variety of coupling forces shall be done. Normally, the maximum coupling forces will give the maximum bending moment and the minimum coupling Figure 11. Deformed structure of segmental forces will cause the maximum deformation. When the lining with swelling loads at one ring lining is opened to build a cross passage or a high locally load has to be applied to a single ring, the bending moments will increase with the decreasing of the possible coupling forces. In such cases a minimum amount of possible coupling forces could be necessary to assure the stability of the whole system. Due to to the high efforts the shown 3D-FEM calculations are not common practice. They should be reserved to cases needed. 6